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Vibration and snapping of a self-contacted beam under prescribed end rotations. (English) Zbl 1476.74049

Summary: An initially straight beam is first bent into a circular configuration. Equal rotation angles of opposite direction are then prescribed at the two ends to bend the beam into self contact and eventually snap. We conduct a vibration analysis based on an Eulerian formulation to determine the natural frequencies and the stability of the bent beam. The theory predicts that the rotated beam snaps sideways at the bifurcation point. In experiments, however, the self-contacted beam passes far beyond the bifurcation point and snaps symmetrically by squeezing itself past the center of the two end points to the other side. By looking into the mode shapes of the unstable mode at the bifurcation point, it is believed that the predicted sideway snapping may be prevented by the sliding friction between the contact surfaces, which is not included in the theory. Instead, the beam snaps in a second unstable mode which involves rolling between contact surfaces. We then propose an imperfection analysis in which the bent beam is slanted to one side by a small angle in its initial configuration. In an experiment with a slant angle of \(3^\circ\), the deformation follows the load-deflection curve of the imperfect model and snaps sideways near the limit point. This experimental result may be considered as an auxuliary evidence of the existence of the bifurcation point and the associated sideway snapping phenomenon in the perfect model.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74M15 Contact in solid mechanics
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[1] Cazzolli, A.; Corso, F. D., Snapping of elastic strips with controlled ends, Int. J. Solid Struct., 162, 285-303 (2019)
[2] Cazzolli, A.; Misseroni, D.; Corso, F. D., Elastica catastrophe machine: theory, design and experiments, J. Mech. Phys. Solid., 136, 103735 (2020)
[3] Chen, J.-S.; Lin, J.-S., Exact critical loads for a pinned half-sine arch under end couples, J. Appl. Mech., 72, 147-148 (2005) · Zbl 1111.74357
[4] Chen, J.-S.; Lin, Y.-C., Vibration and stability of a long heavy elastica on rigid foundation, Int. J. Non Lin. Mech., 50, 11-18 (2013)
[5] Chen, J.-S.; Ro, W.-C., Dynamic response of a shallow arch under end moments, J. Sound Vib., 326, 321-331 (2009)
[6] Chen, J.-S.; Ro, W.-C., Deformations and stability of an elastica subjected to an off-axis point constraint, J. Appl. Mech., 77, Article 031006 pp. (2010)
[7] Chen, J.-S.; Wang, L.-C., Contact between two planar buckled beams pushed together transversely, Int. J. Solid Struct., 199, 181-189 (2020)
[8] Domokos, G.; Fraser, W. B.; Szeberenyi, L., Symmetry-breaking bifurcations of the uplifted elastic strip, Physica D, 185, 67-77 (2003) · Zbl 1098.74533
[9] Evans, M. E.G., The jump of the click beetle (coleopetra, elateridae)-a preliminary study, J. Zool., 167, 319-336 (2009)
[10] Forterre, Y.; Skotheim, J. M.; Dumais, J.; Mahadevan, L., How the venus flytrap snaps, Nature, 433, 421-425 (2005)
[11] Haghpanah, B.; Salari-Sharif, L.; Pourrajab, P.; Hopkins, J., Multistable shape-reconfigurable architected materials, Adv. Mater., 28, 7915-7920 (2016)
[12] Lee, C.-C., Vibration and Snapping of a Self-Contacted Beam under Prescribed End Rotations (2020), National Taiwan University: National Taiwan University Taipei, Taiwan, Master Thesis
[13] Mochiyama, H.; Kinoshita, A.; Takasu, R., Impulse force generator based on snap-through buckling of robotic closed elastica: analysis by quasi-static shape transition simulation, (IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) November 3-7. Tokyo, Japan (2013))
[14] Nadkarni, N.; Arrieta, A. F.; Chong, C.; Kochmann, D. M.; Daraio, C., Unidirectional transition waves in bistable lattices, Phys. Rev. Lett., 116, 244501 (2016)
[15] Ouakad, H. M.; Younis, M. I., On using the dynamic snap-through motion of MEMS initially curved microbeams for filtering applications, J. Sound Vib., 333, 555-568 (2014)
[16] Plaut, R. H., Optimal arch form for stability under end moments, Proceedings of the 18^th Midwestern Mechanics Conference, 87-89 (1983), Iowa City: Iowa City , IA
[17] Plaut, R. H., Snap-through of shallow elastic arches under end moments, J. Appl. Mech., 76, Article 014504 pp. (2009)
[18] Plaut, R. H.; Taylor, R. P.; Dillard, D. A., Postbuckling and vibration of a flexible strip clamped at its ends to a hinged substrate, Int. J. Solid Struct., 41, 859-870 (2004) · Zbl 1075.74534
[19] Plaut, R. H.; Virgin, L. N., Vibration and snap-through of bent elastica strips subjected to end rotations, J. Appl. Mech., 76, Article 041011 pp. (2009)
[20] Raney, J. R.; Nadkarni, N.; Daraio, C.; Kochmann, D. M.; Lewis, J. A.; Bertoldi, K., Stable propagation of mechanical signals in soft media using stored elastic energy, Proc. Natl. Acad. Sci. USA, 113, 9722-9727 (2016)
[21] Ro, W.-C.; Chen, J.-S.; Hong, S.-Y., Vibration and stability of a constrained elastica with variable length, Int. J. Solid Struct., 47, 2143-2154 (2010) · Zbl 1194.74145
[22] Scarselli, G.; Nicassio, F.; Pinto, F.; Ciampa, F.; Iervolino, O.; Meo, M., A novel bistable energy harvesting concept, Smart Mater. Struct., 25, Article 055001 pp. (2016)
[23] Yan, S.-T.; Shen, X.; Jin, Z., Static and dynamic symmetric snap-through of non-uniform shallow arch under a pair of end moments considering critical slowing-down effect, J. Mech. Eng. Sci., 233, 5735-5762 (2019)
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